arXiv research

The study examines stability and isoperimetry of CMC spheres in hyperbolic and spherical manifolds.

The paper characterizes biharmonic maps between spheres using polynomial functions.

Locally flat 2-spheres in $\mathbb{C} P^2$ with knot group $\mathbb{Z}_2$ are ambiently isotopic if homologous.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

Analytic saddle spheres in S^3 are equators.

There is a topological embedding $ι:\mathbb{S}^1\to\mathbb{R}^5$ such that $π_3(\mathbb{R}^5\setminusι(\mathbb{S}^1))=0$. Therefore, no $3$-sphere can be linked with $ι(\mathbb{S}^1)$.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

Authors construct hypertori with constant negative mean curvature in a sphere.

The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.

This paper classifies Möbius homogeneous hypersurfaces in a sphere.

In this paper we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a consequence, we prove that knots in an L-space $\mathbb{Z}$-homology sphere are determined …

In the present study we consider knotted spheres in Euclidean $4$-space $\mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $\mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.

Paper classifies hypersurfaces in a sphere with specific curvature properties.

By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is a $Ω$-stable submanifold with parallel mean curvature, when $Ω$ is the Kähler calibration of rank 4 of $\mathbb{C}^3$.

The study finds infinitely many p-harmonic maps between spheres for specific p and m.

Extends rigidity results for Whitney spheres in higher dimensions.

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.

Study defines hyper-dual spheres and ruled surfaces, proving geometric relationships.

We study generalized Killing spinors on round spheres $\mathbb{S}^n$. We show that on the standard sphere $\mathbb{S}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on $\mathbb{S}^n$ whose associated symmetric endomorphism has at most two eigenva…

New Zoll families of minimal spheres found in spheres and projective spaces.

Minimal surfaces in spheres are classified based on a Ricci-like condition.

The paper constructs four-manifolds with lens space boundaries and explores sphere configurations in $\#^n \mathbb{C} P^2$.

Generalized Thurston's characterization for branched coverings of the 2-sphere.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.

The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact exa…

New method for triharmonic maps to spheres in various dimensions.

The study identifies holomorphic sections on jet spaces of the Riemann sphere.

The paper studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices…

We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds $M^{2n}$, where $n=7$ or $8$, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…

New types of Delaunay hypersurfaces found in spheres.

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

There exists a well known construction which allows to associate with two hyperbolic affine spheres $f_i: M_i^{n_i} \to \mathbb R^{n_i+1}$ a new hyperbolic affine sphere immersion of $I \times M_1 \times M_2$ into $\mathbb R^{n_1+n_2+3}$. In this paper we deal with the inverse problem: how to determine from properties …

Falsehood of Pólya's conjecture for spheres shown.

The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.

Study on minimal two-spheres in complex hyperquadric, proving non-congruence of constant curvature spheres.

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert $\mathbb{Z}/2\mathbb{Z}$-homology sphere or a toroidal irreducible non-Seifert surgery then t…

Study constant and almost constant curvature spheres in hyperbolic space.

Proves spheres with bounded curvatures must contain a unit ball.

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.

Optimal inequality on sphere for convex bodies.

Paper studies equatorial concentration of measure in sphere immersions and submersions.

Explicitly constructs CR regular embeddings of spheres in complex spaces.

Paper studies rigidity of translation surfaces in 3D sphere using quaternionic product.